Compressive sensing lattice dynamics. I. General formalism

Ab initio calculations have been successfully used for evaluating lattice dynamical properties of solids within the (quasi)harmonic approximation (i.e., assuming noninteracting phonons with infinite lifetimes), but it remains difficult to treat anharmonicity in all but the simplest compounds. We detail a systematic information-theory-based approach to deriving ab initio anharmonic force constants: Compressive sensing lattice dynamics (CSLD). The non-negligible terms that are necessary to reproduce the first-principles calculated interatomic forces are automatically selected by minimizing the ${\ell }_1$ norm (sum of absolute values) of the scaled force constants. By using efficient sampling of the configuration space using a modest number of atomic configurations with quasirandom displacements, CSLD is well suited for deriving accurate anharmonic potentials for complex multicomponent compounds with large unit cells. We demonstrate the power and generality of CSLD by calculating the phonon lifetimes and thermal transport properties of type-I Si clathrates.

Figure 1

Taylor expansion of the Morse potential $V\left(r\right)=D_0\left(e^{-2a(r-r_0)}-2e^{-a(r-r_0)}\right)$ with $D_0=1$ and $a=1$.

Figure 2

Log-linear plots of the Frobenius norm of the third-order FCT $\mathrm{\Phi }\left(\alpha \right)$ vs the scaled distance $d_{\alpha }$ for (a) Si, (b) NaCl, and (c) Al.

Figure 3

Fitted pairwise potential $E_{\text{FF}}$ for ${\mathrm{Cu}}_{12}{\mathrm{Sb}}_4{\mathrm{S}}_{13}$.

Figure 4

Comparison of the lattice thermal conductivity of NaCl versus temperature from CSLD-based molecular dynamics simulations in conjunction with the Green-Kubo formula (filled squares) third-order PT $+$ BTE calculations (filled circles), and experimental measurements from Refs. [83, 84, 85, 86] (open symbols). The dashed black line through the PT $+$ BTE results is a guide to the eye.

Figure 5

Crystal structure of type-I Si clathrate with guest atoms. Si and guest atoms are displayed as green and magenta solid spheres, respectively. There are two kinds of polyhedra: Dodecahedral (D) and tetrakaidecahedral (T).

Figure 6

Frobenius norm of (a) second- and (b) third-order FCTs versus interaction distance for ${\mathrm{Si}}_{46}$ and $G_8{\mathrm{Si}}_{46}$ ($G$ = Na, Ba).

Figure 7

Comparison of Frobenius norm of (a) the improper onsite pair FCT ${\mathrm{\Phi }}_{aa}$ and (b) improper onsite third-order $\mathrm{\Phi }aaa$ for ${\mathrm{Si}}_{46}$ and $G_8{\mathrm{Si}}_{46}$ ($G$=Na, Ba). The subscripts on $G$ refer to the two types of cages (dodecahedral and tetrakaidecahedral).

Figure 8

Calculated phonon dispersions for (a) ${\mathrm{Si}}_{46}$, (b) ${\mathrm{Na}}_8{\mathrm{Si}}_{46}$ and (c) ${\mathrm{Ba}}_8{\mathrm{Si}}_{46}$. (d) Calculated lattice thermal conductivities compared to previous theoretical study using effective potential by Schopf et al. [100] (red square), first-principles calculations by Norouzzadeh et al. [101] (red squares) and Euchner et al. [102] (red disks and green triangles), and experimental measurements by Pailhès et al. [91] (green triangle) and Nolas et al. [103] (blue circle). (e) Calculated mode-resolved lifetimes at 300 K compared with experimental measurements performed on ${\mathrm{Ba}}_8{\mathrm{Si}}_{46}$ [91]. (f) Normalized cumulative lattice thermal conductivities at 300 K. (g) Comparison of three-phonon scattering phase space in ${\mathrm{Si}}_{46}$ and $G_8{\mathrm{Si}}_{46}$ ($G$ = Na, Ba). The three-phonon scattering phase space is calculated using ${\mathrm{\Gamma }}_{\lambda {\lambda }^{\prime }{\lambda }^{″}}^{\pm }$ by setting $|V_{\pm \lambda {\lambda }^{\prime }{\lambda }^{″}}^{\left(3\right)}|$ as a constant. Atomic partition ratios of (h) Na and (i) Ba atoms.